一种基于自适应牛顿法的自由边界Grad-Shafranov求解器
An adaptive Newton-based free-boundary Grad-Shafranov solver
AI总结：
针对自由边界Grad-Shafranov方程求解的非线性优化难题，提出结合自适应有限元、形状微积分与AMG预处理的牛顿求解器，可高效收敛并解决常规Picard法失效的泰勒态平衡问题。
摘要：
磁约束装置中的平衡态由洛伦兹力与等离子体压力梯度之间的力平衡产生。在托卡马克这类轴对称构型中，这种平衡由描述极向磁通量的椭圆方程（即众所周知的Grad--Shafranov方程）描述。开发可扩展且精确的自由边界Grad-Shafranov求解器颇具挑战性，因为这是一个完全非线性的优化问题，需要同时求解等离子体外部的磁场线圈电流以控制等离子体形状。本研究中，我们开发了一种结合自适应有限元与预处理策略的基于牛顿法的自由边界Grad-Shafranov求解器。自由边界相互作用需要计算依赖域的非线性形式，其对雅可比矩阵的贡献通过形状微积分实现。该优化问题旨在最小化等离子体边界与指定控制点之间的距离，同时满足两个重要约束：一是Grad-Shafranov方程的非线性有限元离散，二是包含非局部耦合项的总等离子体电流约束。线性系统通过块分解求解，子块椭圆算子则调用AMG（代数多重网格）求解。本工作的独特贡献包括全局约束的处理、预处理策略、非局部重构以及自适应有限元的实现。研究发现，所得牛顿求解器鲁棒性强，仅需少量迭代即可成功将非线性残差降至1e-6及以下，同时能解决常规基于Picard的求解器无法收敛的难题——求解泰勒态平衡。
英文摘要：
Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad--Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad--Shafranov solver, since it is a fully nonlinear optimization problem that simultaneously solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad--Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two non-trivial constraints, which correspond to the nonlinear finite element discretization of the Grad--Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. It is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.
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